Exponent rules — multiplication, division, power of power, zero and negative exponents

Question

Simplify: (a) $2^3 \times 2^5$, (b) $5^7 \div 5^4$, (c) $(3^2)^4$, (d) $7^0$, (e) $4^{-2}$. State the rule used in each.

Solution — Step by Step

Why This Works

graph TD
    A["Which exponent rule?"] --> B["Same base, multiplication?"]
    B -->|Yes| C["ADD exponents: aᵐ × aⁿ = aᵐ⁺ⁿ"]
    A --> D["Same base, division?"]
    D -->|Yes| E["SUBTRACT exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ"]
    A --> F["Power of a power?"]
    F -->|Yes| G["MULTIPLY exponents: aᵐⁿ = aᵐˣⁿ"]
    A --> H["Exponent is 0?"]
    H -->|Yes| I["Answer is 1"]
    A --> J["Exponent is negative?"]
    J -->|Yes| K["Flip to denominator: a⁻ⁿ = 1/aⁿ"]

Exponents are shorthand for repeated multiplication. $2^3 = 2 \times 2 \times 2$ (three 2s multiplied). So $2^3 \times 2^5 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2) = 2^8$ — we just counted all the 2s, which gives us the “add exponents” rule.

The zero exponent rule follows from division: $a^n \div a^n = a^{n-n} = a^0$, but also $a^n \div a^n = 1$. So $a^0 = 1$.

Negative exponents extend the pattern: $a^2, a^1, a^0, a^{-1}, a^{-2}$ — each step divides by $a$. So $a^{-1} = 1/a$ and $a^{-2} = 1/a^2$.

Alternative Method

Common Mistake